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Properties

Label 2-3e5-243.103-c1-0-8
Degree 22
Conductor 243243
Sign 0.976−0.214i0.976 - 0.214i
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.83 − 0.509i)2-s + (1.22 + 1.22i)3-s + (1.37 + 0.832i)4-s + (3.83 − 0.148i)5-s + (−1.61 − 2.87i)6-s + (0.365 + 0.0571i)7-s + (0.508 + 0.538i)8-s + (−0.0160 + 2.99i)9-s + (−7.09 − 1.68i)10-s + (−0.814 − 5.96i)11-s + (0.662 + 2.70i)12-s + (−3.43 + 5.00i)13-s + (−0.639 − 0.290i)14-s + (4.86 + 4.52i)15-s + (−2.15 − 4.09i)16-s + (3.18 − 1.60i)17-s + ⋯
L(s)  = 1  + (−1.29 − 0.360i)2-s + (0.705 + 0.708i)3-s + (0.689 + 0.416i)4-s + (1.71 − 0.0665i)5-s + (−0.657 − 1.17i)6-s + (0.138 + 0.0215i)7-s + (0.179 + 0.190i)8-s + (−0.00535 + 0.999i)9-s + (−2.24 − 0.531i)10-s + (−0.245 − 1.79i)11-s + (0.191 + 0.782i)12-s + (−0.951 + 1.38i)13-s + (−0.170 − 0.0776i)14-s + (1.25 + 1.16i)15-s + (−0.539 − 1.02i)16-s + (0.772 − 0.388i)17-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.976−0.214i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.976−0.214i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.976−0.214i0.976 - 0.214i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(103,⋅)\chi_{243} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.976−0.214i)(2,\ 243,\ (\ :1/2),\ 0.976 - 0.214i)

Particular Values

L(1)L(1) ≈\approx 1.02997+0.111722i1.02997 + 0.111722i
L(12)L(\frac12) ≈\approx 1.02997+0.111722i1.02997 + 0.111722i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−1.22−1.22i)T 1 + (-1.22 - 1.22i)T
good2 1+(1.83+0.509i)T+(1.71+1.03i)T2 1 + (1.83 + 0.509i)T + (1.71 + 1.03i)T^{2}
5 1+(−3.83+0.148i)T+(4.98−0.387i)T2 1 + (-3.83 + 0.148i)T + (4.98 - 0.387i)T^{2}
7 1+(−0.365−0.0571i)T+(6.66+2.13i)T2 1 + (-0.365 - 0.0571i)T + (6.66 + 2.13i)T^{2}
11 1+(0.814+5.96i)T+(−10.5+2.94i)T2 1 + (0.814 + 5.96i)T + (-10.5 + 2.94i)T^{2}
13 1+(3.43−5.00i)T+(−4.68−12.1i)T2 1 + (3.43 - 5.00i)T + (-4.68 - 12.1i)T^{2}
17 1+(−3.18+1.60i)T+(10.1−13.6i)T2 1 + (-3.18 + 1.60i)T + (10.1 - 13.6i)T^{2}
19 1+(−0.159−2.73i)T+(−18.8+2.20i)T2 1 + (-0.159 - 2.73i)T + (-18.8 + 2.20i)T^{2}
23 1+(0.230−0.597i)T+(−17.0−15.4i)T2 1 + (0.230 - 0.597i)T + (-17.0 - 15.4i)T^{2}
29 1+(−0.792−0.566i)T+(9.38+27.4i)T2 1 + (-0.792 - 0.566i)T + (9.38 + 27.4i)T^{2}
31 1+(−3.51+4.03i)T+(−4.19−30.7i)T2 1 + (-3.51 + 4.03i)T + (-4.19 - 30.7i)T^{2}
37 1+(3.14+4.22i)T+(−10.6+35.4i)T2 1 + (3.14 + 4.22i)T + (-10.6 + 35.4i)T^{2}
41 1+(0.857−3.32i)T+(−35.8−19.8i)T2 1 + (0.857 - 3.32i)T + (-35.8 - 19.8i)T^{2}
43 1+(−3.41−3.09i)T+(4.16+42.7i)T2 1 + (-3.41 - 3.09i)T + (4.16 + 42.7i)T^{2}
47 1+(2.84+3.25i)T+(−6.36+46.5i)T2 1 + (2.84 + 3.25i)T + (-6.36 + 46.5i)T^{2}
53 1+(0.566−3.21i)T+(−49.8−18.1i)T2 1 + (0.566 - 3.21i)T + (-49.8 - 18.1i)T^{2}
59 1+(1.88−0.768i)T+(42.1−41.3i)T2 1 + (1.88 - 0.768i)T + (42.1 - 41.3i)T^{2}
61 1+(12.5−7.55i)T+(28.4−53.9i)T2 1 + (12.5 - 7.55i)T + (28.4 - 53.9i)T^{2}
67 1+(−6.29+4.49i)T+(21.6−63.3i)T2 1 + (-6.29 + 4.49i)T + (21.6 - 63.3i)T^{2}
71 1+(2.46+8.24i)T+(−59.3+39.0i)T2 1 + (2.46 + 8.24i)T + (-59.3 + 39.0i)T^{2}
73 1+(6.59−1.56i)T+(65.2−32.7i)T2 1 + (6.59 - 1.56i)T + (65.2 - 32.7i)T^{2}
79 1+(−2.56+2.51i)T+(1.53−78.9i)T2 1 + (-2.56 + 2.51i)T + (1.53 - 78.9i)T^{2}
83 1+(3.79+14.7i)T+(−72.6+40.0i)T2 1 + (3.79 + 14.7i)T + (-72.6 + 40.0i)T^{2}
89 1+(−1.54+5.15i)T+(−74.3−48.9i)T2 1 + (-1.54 + 5.15i)T + (-74.3 - 48.9i)T^{2}
97 1+(−12.6−0.491i)T+(96.7+7.51i)T2 1 + (-12.6 - 0.491i)T + (96.7 + 7.51i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.70125728845483094432942226737, −10.67988914453321110546972918961, −9.960623033504755833355071575333, −9.331087809088576834545953659302, −8.723960214643556489890935593606, −7.66032838832732826768172577374, −6.01757938571794255780161478084, −4.90362268289885828360075691966, −2.89214731814709362922088217142, −1.74348449633146173793495728005, 1.50431494722307919795784407147, 2.61136326754719227979749585475, 5.06520099139419757479017123994, 6.48232464858554348247261287121, 7.32163504698286396040014925662, 8.136131331279243302595917577193, 9.278532967928895857893155769636, 9.933867138241574732684099312726, 10.35692356368972010541491823263, 12.46272208292836898737940331811

Graph of the ZZ-function along the critical line